Cramer’s Rule for a 3×3 System (with Three Variables)

In our previous lesson, we studied how to use Cramer’s Rule with two variables. Our goal here is to expand the application of Cramer’s Rule to three variables usually in terms of x, y, and z. I will go over five (5) worked examples to help you get familiar with this concept.

To do well on this topic, you need to have an idea how to find the determinant of a 3×3 matrix. So, this is what we are going to do first. Ready?

Formula to Find the Determinant of a 3×3 Matrix

Given a 3×3 matrix

Its determinant can be calculated using the following formula.

Let’s do a quick example on this…

Find the determinant of matrix

Solution: Make sure that you follow the formula on how to find the determinant of a 3×3 matrix carefully, as shown above. More so, don’t rush when you perform the required arithmetic operations in every step. This is where common errors usually occur, but it can be prevented. When you do it right, your solution should be similar to the one below.

Now, it’s time to go over the procedure on how to use Cramer’s Rule in a linear system involving three variables.

Cramer’s Rules for Systems of Linear Equations with Three Variables

Given a linear system

Labeling each of the four matrices

coefficient matrix:

X – matrix:

Y – matrix:

Z – matrix:

To solve for x:

To solve for y:

To solve for z:

Things to observe from the setup above:

1) The coefficients of variables x, y and z make use of subscripted a, b, and c, respectively. While the constant terms use subscripted d.

2) The denominators to find the values of x , y and z are all the same which is the determinant of the coefficient matrix (coefficients coming from the columns of x, y, and z).

3) To solve for x, the coefficients of x-column is replaced by the constant column (in red).

4) To solve for y, the coefficients of y-column is replaced by the constant column (in red).

5) In the same manner, to solve for z, the coefficients of z-column is replaced by the constant column (in red).

Examples of How to Solve Systems of Linear Equations with Three Variables using Cramer’s Rule

Example 1: Solve the system with three variables by Cramer’s Rule

From the given system of linear equations, I will construct the four matrices that will be used to solve for the values of x, y, and z.

Use the guide above to correctly setup these special matrices.

coefficient matrix

X – matrix

Y – matrix

Z – matrix

Next, I will solve for the determinant of each matrix. To do this, I can manually solve the determinant of each matrix on paper using the formula provided above. It can be tedious, but it’s okay since good math skills are developed by doing lots of problems.

To check your work, you may use the 3×3 determinant calculator from Wolfram Alpha. Type in the entries then click calculate.

The values of the determinants are listed below.

Determinants of each matrix

The final answers or solutions are easily computed or calculated once all the required determinants are found.

Solved values for x, y, and z

The final answer written in point notation is (x, y, z) = (−1, 1, −2).

Example 2: Solve the system with three variables by Cramer’s Rule

I actually consider the coefficient matrix as the “primary” matrix because the other three matrices are derived from it. For instance, the x-matrix is just the “primary” matrix with the x-column replaced by the constant column (in red). You can observe that same pattern is applied in constructing the other matrices: y and z.

coefficient matrix

X – matrix

Y – matrix

Z – matrix

After solving the determinant of each matrix, I have them all written down.

Determinants of each matrix

The values for x, y and z are calculated as follows. Notice that x is obtained by taking the determinant of the x-matrix divided by the determinant of the coefficient matrix. This rule holds for the rest.

Solved values for x, y, and z

Our final answer is (x, y, z) = (−4, 2, 1).

Example 3: Solve the system with three variables by Cramer’s Rule

This problem is much easier than the first two examples because of the presence of zero entries in the x, y, and constant columns. Do you see it? When we have zero entries in a matrix, the calculation of its determination is dramatically simplified.

In fact, as you increase the number of zeroes in a square matrix, the work done to find its determinant is greatly reduced.

Here are the matrices extracted from the system of linear equations.

coefficient matrix

X – matrix

Y – matrix

Z – matrix

Solving for their determinants, I got the following values…

Determinants of each matrix

This leads us to easily setup and calculates the final answers.

Solved values for x, y, and z

The final answer is (x, y, z) = (−1, 6, 1).

Example 4: Solve the system with three variables by Cramer’s Rule

Write down the four special matrices.

coefficient matrix

X – matrix

Y – matrix

Z – matrix

Evaluate each matrix to find its determinant.

Determinants of each matrix

Use the Cramer’s Rule to get the following solutions.

Solved values for x, y, and z

The final answer is (x, y, z)= (−1, 2, 0).

Example 5: Solve the system with three variables by Cramer’s Rule

Let’s do one final example! I hope that at this point, you have had enough practice on how to solve systems with three variables using Cramer’s Rule.

I suggest that you solve this on paper first and then come back to compare your answer.